The Nilpotency Class of Fitting Subgroups of Groups with Basis Property (original) (raw)

Nilpotency: A Characterization Of The Finite p-Groups

Journal of Mathematics , 2017

Abstract As parts of the characterizations of the finite p-groups is the fact that every finite p-group is NILPOTENT. Hence, there exists a derived series (Lower Central) which terminates at e after a finite number of steps. Suppose that G is a p-group of class at least m ≥ 3. Then L m-1G is abelian and hence G possesses a characteristic abelian subgroup which is not contained in Z(G). If L 3(G) = 1 such that pm is the highest order of an element of G/L2 (G) (where G is any p-group) then no element of L2(G) has an order higher than pm. [1]

On a Class of Generalized Nilpotent Groups

Journal of Algebra, 2002

We explore the class of generalized nilpotent groups in the universe c of all radical locally finite groups satisfying min-p for every prime p. We obtain that this class is the natural generalization of the class of finite nilpotent groups from the finite universe to the universe c. Moreover, the structure of-groups is determined explicitly. It is also shown that is a subgroup-closed c-formation and that in every c-group the Fitting subgroup is the unique maximal normal-subgroup.

Some necessary and sufficient conditions for p-nilpotence of finite groups

Bulletin of the Australian Mathematical Society, 2003

The purpose of this paper is to give some necessary and sufficient conditions for p-nilpotent groups. We extend some results, including the well-known theorems of Burnside and Frobenius as well as some very recent theorems. We also apply our results to determine the structure of some finite groups in terms of formation theory.

On Finite-By-Nilpotent Groups

Glasgow Mathematical Journal

Abstarct Let γ n = [x1,…,x n ] be the nth lower central word. Denote by X n the set of γ n -values in a group G and suppose that there is a number m such that ∣gXn∣lem|{g^{{X_n}}}| \le m∣gXn∣lem for each g ∈ G. We prove that γn+1(G) has finite (m, n) -bounded order. This generalizes the much-celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite.

A note on p-nilpotence and solvability of finite groups

Journal of Algebra, 2009

In this note, we first give some examples to show that some hypotheses of some well-known results for a group G to be pnilpotent, solvable and supersolvable are essential and cannot be removed. Second, we give some generalizations of two theorems in [A. Ballester-Bolinches, X. Guo, Some results on p-nilpotence and solubility of finite groups, J. Algebra 228 (2000) 491-496].

On subgroups generated by small classes in finite groups

2009

Let GGG be a finite group and M(G)M(G)M(G) be the subgroup of GGG generated by all non-central elements of GGG that lie in the conjugacy classes of the smallest size. Recently several results have been proved regarding the nilpotency class of M(G)M(G)M(G) and F(M(G))F(M(G))F(M(G)), where F(M(G))F(M(G))F(M(G)) denotes the Fitting subgroup of M(G)M(G)M(G). We prove some conditional results regarding the nilpotency class of M(G)M(G)M(G).

On a combinatorial problem in group theory

Israel journal of mathematics, 1993

Let n be a positive integer or infinity (denote ∞). We denote by W * (n) the class of groups G such that, for every subset X of G of cardinality n + 1, there exist a positive integer k, and a subset X 0 ⊆ X, with 2 ≤ |X 0 | ≤ n + 1 and a function f : {0, 1, 2,. .. , k} −→ X0, with f (0) = f (1) and non-zero integers t0, t1,. .. , t k such that [x t 0 0 , x t 1 1 ,. .. , x t k k ] = 1, where xi := f (i), i = 0,. .. , k, and xj ∈ H whenever x t j j ∈ H, for some subgroup H = D x t j j E of G. If the integer k is fixed for every subset X we obtain the class W * k (n). Here we prove that (1) Let G ∈ W * (n), n a positive integer, be a finite group, p > n a prime divisor of the order of G, P a Sylow p-subgroup of G. Then there exists a normal subgroup K of G such that G = P × K. (2) A finitely generated soluble group has the property W * (∞) if and only if it is finite-by-nilpotent. (3) Let G ∈ W * k (∞) be a finitely generated soluble group, then G is finite-by-(nilpotent of k-bounded class).